Necklaces, subset sums, and cyclic permutations

Abstract

It is a well known that, for odd $n$, the number of subsets of $\{1,2,\dots,n\}$ the sum of whose elements is divisible by $n$ equals the number of binary necklaces of length $n$. In this paper generalize this result in two directions. On the one hand, we introduce a parameter $r$ so that requiring the subset sums to be congruent to $r$ modulo $n$ translates into imposing some periodicity conditions on the necklaces. On the other hand, we refine these relations by the size $k$ of the subset, showing that it matches the number of ones in the necklace. We describe the precise conditions on $n$, $k$ and $r$ for which the equalities hold. We also extend some of our formulas to $q$-ary necklaces. The classical results correspond to the case $r=0$. When $r=1$, our identity is related to a conjecture of Baker et al. connecting subsets the sum of whose elements is congruent to $1$ modulo $n$ and unimodal permutations which consist of one cycle. We prove this conjecture using generating functions. Finding bijective proofs of most of our identities remains an open problem.